#367430

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^c=di

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, 274626);