#7631961

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	#35728	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5350 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539157);