#234595

order	512
nilpotency class	3
isoclinism	isoclinic class of #232273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35000	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141791);