C₂×#12302 #484573

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12302

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 138 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);(3,4,9,14,10,16,8,15)(5,17,21,38,22,46,20,45)(6,23,25,12,26,41,31,37)(7,27,29,13,30,42,32,40)(11,28,43,19,35,51,34,52)(18,47,49,44,50,65,53,62)(24,39,58,55,59,36,57,60)(33,61,54,56,64,48,63,66);(3,5,10,22)(4,11,16,35)(6,18,26,50)(7,28,30,51)(8,21,9,20)(12,33,37,64)(13,38,40,45)(14,34,15,43)(17,27,46,42)(19,29,52,32)(23,54,41,63)(24,56,59,66)(25,53,31,49)(36,47,39,65)(44,60,62,55)(48,58,61,57);(3,6)(4,12)(5,18)(7,24)(8,31)(9,25)(10,26)(11,33)(13,39)(14,41)(15,23)(16,37)(17,44)(19,48)(20,53)(21,49)(22,50)(27,60)(28,56)(29,58)(30,59)(32,57)(34,63)(35,64)(36,40)(38,65)(42,55)(43,54)(45,47)(46,62)(51,66)(52,61)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391769);