#343931

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
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

512 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)(3,4)(5,7)(6,8)(9,10,14,16,12,11,13,15)(18,23)(19,27)(25,26)(29,30);(1,3)(2,6)(4,7)(5,8)(10,11)(15,16)(17,18,21,23)(19,28,27,32)(20,26,22,25)(24,30,31,29);(1,4,5,6)(2,3,7,8)(9,11,12,10)(13,15,14,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,31)(25,28)(26,32);(2,7)(3,8)(9,10,12,11)(13,16,14,15)(17,20,21,22)(18,25,23,26)(19,29,27,30)(24,28,31,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=ei,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 251127);