#7629578

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

5635 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)(11,15)(12,14)(13,16)(17,18,23,22)(19,20,21,24);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19,23,21)(18,24,22,20);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,16,15)(17,20,23,24)(18,19,22,21);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(17,21,23,19)(18,24,22,20);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,16,15)(17,22,23,18)(19,20,21,24);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,c^b=ci,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257471, #72257535, #72264282
maximal normal subgroup of	#72257471, #72257535, #72264282
maximal quotient of	#72264325

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536774);