#7639629

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

5407 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,3,5,4)(2,6,8,7)(9,10,14,11)(12,16,13,15)(17,18)(19,23)(20,22)(21,24);(1,3,5,4)(2,7,8,6)(9,11,14,10)(12,16,13,15)(17,19)(18,21)(20,24)(22,23);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,15,11,16)(18,22)(21,23);(1,3,5,4)(2,7,8,6)(9,13,14,12)(10,15,11,16)(19,24)(21,23);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,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.

Relationships

maximal subgroup of	#72303642
maximal normal subgroup of	#72303642
maximal quotient of	#72303657, #72303665

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7546825);