#6463771

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 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)(15,25)(16,28)(17,22)(18,23)(19,24)(20,26)(21,27);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,16,19,27,17,26)(14,20,22,21,24,28,23,25);(6,9)(10,11)(13,16,18,27,19,26,17,15)(14,21,22,28,24,25,23,20);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,16,27,26)(20,25,28,21);(13,17,19,18)(14,22,24,23)(15,26,27,16)(20,21,28,25)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,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=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370967);