#453932

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

811 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)(3,7,8,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24);(1,3,5,8)(2,6,4,7)(9,11)(10,14,16,17)(12,21,22,18)(13,19)(15,24)(20,23);(1,2,5,4)(3,6,8,7)(14,17)(23,24);(1,4,5,2)(3,6,8,7)(9,12)(10,15,16,20)(11,18)(13,22)(14,23,17,24)(19,21)

Polycyclic presentation:

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

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, 361128);