#7674748

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581944);