#184709

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,d^a=dehi,c^b=cei,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91905);