#330692

order	512
nilpotency class	3
isoclinism	isoclinic class of #329707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 237888);