#7633174

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	#35165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5384 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)(3,8,4,7)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,20,24,23)(21,25,30,29)(27,28,32,31);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28);(1,4,6,3)(2,8,5,7)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31);(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=g,f^2=i,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.

Relationships

maximal subgroup of	#72342290, #72342315, #72342332
maximal normal subgroup of	#72342290, #72342315, #72342332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540370);