#7639104

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4792 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)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31);(9,11)(10,13)(12,16)(14,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(10,14)(13,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31);(11,16)(13,15)(20,28)(23,25)(27,32)(30,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 7546300);