#361373

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323061
maximal normal subgroup of	#72323061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268569);