#6465997

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

2140 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)(5,6,8,10)(7,11,12,9);(2,4)(13,14)(15,24)(16,20)(17,22)(18,27)(19,23)(21,26)(25,28);(5,7,8,12)(6,9,10,11)(13,15,18,16,19,26,17,25)(14,20,22,24,23,28,27,21);(13,16,18,26,19,25,17,15)(14,21,22,20,23,24,27,28);(13,17,19,18)(14,22,23,27)(15,25,26,16)(20,24,28,21)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,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, 6373193);