#297349

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#1600	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fh,d^2=ehi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204545);