#6445046

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24270	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,20)(18,23)(19,21)(22,26)(24,27)(25,28);(1,2)(3,4)(6,9)(10,12)(16,24)(20,27)(22,28)(25,26);(2,4)(13,15,17,14,19,23,18,21)(16,22,26,20,24,28,25,27);(2,4)(13,16,19,24)(14,20,21,27)(15,22,23,28)(17,26,18,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,15,21,23)(16,25,24,26)(20,28,27,22)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352242);