C₂²×#359 #33383

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#359

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	#102
prime radical	C₂²×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 231 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#359, S₄×#359, #72342733
maximal subgroup of	#146728, #146729, #146754, …
maximal normal subgroup of	#146728, #146729, #146754, …
maximal quotient of	C₂²×#1331, C₂²×#1332, C₂×#11591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26371);