C₂²×#508 #62628

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#508

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator, Frattini, cosocle	C₈	C₂⁵
socle	C₂³	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1059 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 36 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#508, S₄×#508, #72342730, …
maximal subgroup of	#147900, #147911, #147916, …
maximal normal subgroup of	#147900, #147911, #147916, …
maximal quotient of	C₂²×#2138, C₂²×#2158, C₂²×C₄×D₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55616);