#415634

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15203
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 322830);