#6467103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284963, #72285302
maximal normal subgroup of	#72284963, #72285302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374299);