C₂×#2222 #33615

order	256
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2222

Subgroups

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(1,2);(3,4,10,16)(5,17,23,45)(6,24,27,41)(7,28,30,39)(8,15,9,14)(11,32,36,20)(12,31,38,26)(13,29,40,18)(19,48,51,65)(21,44,22,43)(25,56,57,37)(33,60,61,66)(34,52,35,46)(42,53,64,50)(47,55,58,63)(49,54,59,62);(3,5,18,46,9,22,30,32,10,23,29,52,8,21,7,20)(4,11,28,44,15,35,40,45,16,36,39,43,14,34,13,17)(6,19,47,66,26,50,57,59,27,51,58,60,31,53,25,49)(12,33,55,48,24,54,56,64,38,61,63,65,41,62,37,42);(3,6)(4,12)(5,19)(7,25)(8,31)(9,26)(10,27)(11,33)(13,37)(14,41)(15,24)(16,38)(17,42)(18,47)(20,49)(21,53)(22,50)(23,51)(28,55)(29,58)(30,57)(32,59)(34,62)(35,54)(36,61)(39,63)(40,56)(43,65)(44,48)(45,64)(46,66)(52,60)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#152506, #152515, #152639, …
maximal normal subgroup of	#152506, #152515, #152639, …
maximal quotient of	C₂×#12431, C₂×#12434, #490475, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26603);