#6373232

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2294 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,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, 6280428);