#517956

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

2134 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fgh,d^2=i,e^2=g,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, 425152);