#306406

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 213602);