C₂×#2151 #33396

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2151

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂³
commutator	C₂²	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 271 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(4,6)(7,8,10,14,11,15,20,26,12,16,21,27,22,28,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(3,4,5,6)(7,9)(8,13)(10,18)(11,19)(12,17)(14,24)(15,25)(16,23)(20,31)(21,29)(22,30)(26,35)(27,33)(28,34)(32,37)(36,38);(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#7239, C₂×#7240, #146957, …
maximal normal subgroup of	C₂×#7239, C₂×#7240, #146957, …
maximal quotient of	C₂×#11997, C₂×#12005, C₂×#12014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26384);