#394853

order	512
nilpotency class	3
isoclinism	isoclinic class of #394064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281836, #72281865, #72282277
maximal normal subgroup of	#72281836, #72281865, #72282277

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302049);