#493884

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1910 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281395, #72281991, #72282017
maximal normal subgroup of	#72281395, #72281991, #72282017
maximal quotient of	#72293115

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401080);