#7685600

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

4545 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)(3,7)(4,8)(5,6)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32);(9,11,13,16)(10,14,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32);(2,6)(3,8)(9,10,13,12)(11,14,16,15)(18,23)(19,26)(25,31)(27,32);(1,3,5,8)(2,7,6,4)(17,19,22,26)(18,24,23,21)(20,27,29,32)(25,28,31,30);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,28)(26,32);(17,21,22,24)(18,26,23,19)(20,28,29,30)(25,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=i,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, 7592796);