#6373208

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280404);