#323605

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230801);