#409650

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265390, #72307328, #72307377
maximal normal subgroup of	#72265390, #72307328, #72307377

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316846);