#6621069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528265);