#6457369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364565);