#453823

order	512
nilpotency class	3
isoclinism	isoclinic class of #427387
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, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

977 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)(12,20,21,15)(13,16)(14,19)(18,23,24,22);(1,3,5,8)(2,6,4,7)(9,11)(10,14,16,17)(12,18,21,24)(13,19)(15,22)(20,23);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,18,19,24)(13,21)(14,22,17,23)(16,20)

Polycyclic presentation:

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

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, 361019);