#6532602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500010
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310812
maximal normal subgroup of	#72310812

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439798);