#7718100

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	#43531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4615 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7625296);