#7632000

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	#35091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5144 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,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, 7539196);