#452152

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,d^2=f,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359348);