#323604

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

618 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)(17,18,23,31,21,24,22,25)(19,26,32,29,27,20,28,30);(1,3,5,4)(2,6,8,7)(9,11,14,16,13,10,12,15)(17,19)(18,20)(21,27)(22,28)(23,32)(24,26)(25,29)(30,31);(9,11,13,10)(12,16,14,15)(17,20,21,26)(18,19,24,27)(22,30,23,29)(25,32,31,28);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,15,11,16)(18,24)(19,27)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=h,d^2=ehi,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, 230800);