#7699118

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	#39128	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5001 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=h,e^2=gh,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, 7606314);