#7674571

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581767);