#411909

order	512
nilpotency class	3
isoclinism	isoclinic class of #394061
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258127, #72258132
maximal normal subgroup of	#72258127, #72258132

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319105);