#354049

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3490 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239777
maximal normal subgroup of	#72239777
maximal quotient of	#72239585, #72239587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261245);