#303270

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210466);