#7638516

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#39469	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5180 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545712);