#7639826

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4835 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)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(2,5)(6,8)(11,15)(14,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,28)(29,32);(9,10,12,13)(11,14,15,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(9,11,12,15)(10,14,13,16)(19,26)(22,30)(25,27)(29,31);(10,13)(14,16)(20,28)(23,32)(25,27)(29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7547022);