#123656

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#383
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=cf,g^b=gi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235504
maximal normal subgroup of	#72235504
maximal quotient of	#72235546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30852);