#7638585

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

5892 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)(3,8)(4,6)(5,7)(10,14)(11,12);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(18,21)(22,24);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,21)(23,24);(9,12,13,11)(10,15,14,16);(18,21)(22,24);(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,f^2=hi,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	#72257891, #72304334, #72304437, …
maximal normal subgroup of	#72257891, #72304334, #72304437, …
maximal quotient of	#72240757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545781);