#7716008

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	#39358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5092 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)(4,7,8,6)(10,14)(15,16)(17,18,22,24)(19,25,26,23)(20,21,29,30)(27,31,32,28);(9,10,13,14)(11,15,12,16)(17,19,22,26)(18,23,24,25)(20,27,29,32)(21,28,30,31);(2,3)(6,7)(18,24)(19,26)(20,29)(28,31);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,21)(19,27)(22,29)(23,28)(24,30)(25,31)(26,32);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,32)(30,31);(17,21)(18,20)(19,28)(22,30)(23,27)(24,29)(25,32)(26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=gh,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, 7623204);