#395677

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2534 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239135, #72239144, #72239146
maximal normal subgroup of	#72239135, #72239144, #72239146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302873);