#417387

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ef,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324583);