#453771

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=fg,c^2=efg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=dei,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360967);