#7707981

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

5422 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)(11,16)(12,15)(13,14)(17,18,20,21)(19,22,24,23);(9,11)(10,15)(12,13)(14,16)(17,18)(19,23)(20,21)(22,24);(10,14)(15,16)(18,21)(22,23);(9,12,13,11)(10,15,14,16)(17,19,20,24)(18,22,21,23);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|b^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303831
maximal normal subgroup of	#72303831
maximal quotient of	#72304110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615177);