#7117815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#23963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

2^9|a^2=fghi,c^2=ghi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

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, 7025011);