#6370415

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2388 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,d^a=dfi,c^b=cf,d^b=di,d^c=di,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277611);