#312713

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(17,18,22,24)(19,25,27,32)(20,29,28,23)(21,31,30,26);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,15)(14,16)(17,19,21,28,22,27,30,20)(18,23,26,25,24,29,31,32);(2,6)(5,7)(10,13)(11,16)(14,15)(18,24)(19,27)(20,28)(26,31);(2,6)(5,7)(17,20,30,27,22,28,21,19)(18,25,31,23,24,32,26,29)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ef,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219909);