#398900

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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)(6,7)(9,10,13,15)(11,16,12,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(5,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(5,6,8,7)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(2,4)(5,7,8,6)(9,12,13,11)(10,14,15,16)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=fi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi

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, 306096);