#517638

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

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, 424834);