#6598567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574008
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	#36410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 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)(13,14,18,23)(15,24,27,28)(16,21,17,22)(19,25,20,26);(5,6,8,9)(7,10,11,12)(13,15,16,26,18,27,17,25)(14,19,22,24,23,20,21,28);(2,4)(6,9)(7,11)(13,15,16,26,18,27,17,25)(14,20,22,28,23,19,21,24);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,24,20,28);(6,9)(7,11)(13,16,18,17)(14,21,23,22)(15,26,27,25)(19,28,20,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308805, #72308846
maximal normal subgroup of	#72308805, #72308846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505763);