#382538

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2378 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 330 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215054, #72216189, #72217034
maximal normal subgroup of	#72215054, #72216189, #72217034
maximal quotient of	#72214941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289734);