#7689128

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596324);