#6443079

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5490 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312984, #72313010, #72313110
maximal normal subgroup of	#72312984, #72313010, #72313110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350275);