#420201

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307852, #72307861, #72307883
maximal normal subgroup of	#72307852, #72307861, #72307883

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327397);