#7671258

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	#38362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4850 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)(9,10,13,16)(11,14,12,15)(17,18,23,20)(19,27,28,24)(21,25,29,30)(22,31,32,26);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,28)(18,24,20,27)(21,22,29,32)(25,26,30,31);(2,7)(4,8)(19,28)(21,29)(24,27)(25,30);(1,4,6,8)(2,3,7,5)(9,11,13,12)(10,14,16,15)(17,20,23,18)(19,27,28,24)(21,30,29,25)(22,31,32,26);(1,3,6,5)(2,4,7,8)(9,12,13,11)(10,15,16,14)(17,21,23,29)(18,25,20,30)(19,22,28,32)(24,26,27,31);(1,5,6,3)(2,8,7,4)(17,22)(18,26)(19,29)(20,31)(21,28)(23,32)(24,30)(25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=gh,f^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, 7578454);