#7184591

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	#44912	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,23,24,22,27,21,26)(19,28,32,31,29,25,30,20);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(18,24)(20,25)(21,23)(26,27)(28,31)(30,32);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,25,27,28)(20,26,31,24)(21,30,23,32);(10,13)(14,16)(17,20,23,28,22,31,21,25)(18,19,24,32,27,29,26,30);(9,11,12,15)(10,14,13,16)(19,29)(20,31)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=ghi,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7091787);