#6357318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357317
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214074, #72338441, #72338444
maximal normal subgroup of	#72214074, #72338441, #72338444
maximal quotient of	#72224782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264514);