#6996978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6904174);