#6389946

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280879, #72281030, #72282542
maximal normal subgroup of	#72280879, #72281030, #72282542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6297142);