#6718672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#28364	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6625868);