#7638187

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

5804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,24,21);(1,3,5,4)(2,6,7,8)(17,19,20,24)(18,21,22,23);(9,11)(10,13)(12,16)(14,15)(18,22)(19,24);(10,14)(13,15)(18,22)(21,23);(11,16)(13,15);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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.

Relationships

maximal subgroup of	#72305479
maximal normal subgroup of	#72305479
maximal quotient of	#72306261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545383);