#7709924

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5013 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)(11,15)(12,14)(13,16)(17,18,23,20)(19,27,28,24)(21,25,30,31)(22,32,29,26);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,27)(21,29)(22,30)(23,28)(25,32)(26,31);(10,14)(13,15)(19,28)(21,30)(24,27)(25,31);(1,2,5,4)(3,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,18)(19,27,28,24)(21,31,30,25)(22,32,29,26);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,21,23,30)(18,25,20,31)(19,29,28,22)(24,32,27,26);(1,4,5,2)(3,7,8,6)(17,22,23,29)(18,26,20,32)(19,30,28,21)(24,31,27,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239255
maximal normal subgroup of	#72239255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617120);