#7670917

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	#35365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4869 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578113);