#7632442

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	#36273	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=hi,e^2=gi,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, 7539638);