#453931

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, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

807 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)(15,20)(18,24);(1,3,5,8)(2,6,4,7)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,4,5,2)(3,6,8,7)(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24)

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,c^b=cg,d^b=de,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, 361127);