#378145

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226912, #72226917, #72226944
maximal normal subgroup of	#72226912, #72226917, #72226944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285341);