#6587647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25116	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284832, #72284905
maximal normal subgroup of	#72284832, #72284905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494843);