#6441735

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#34737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307559, #72307827, #72307900
maximal normal subgroup of	#72307559, #72307827, #72307900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348931);