#544372

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

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

3534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(1,2)(3,4)(5,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(2,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39);(11,18)(14,15)(19,24)(21,22)(27,34)(30,38)(33,40)(37,39);(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451568);