#7694899

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	#37694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5024 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602095);