#6371970

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280597, #72281079, #72282332
maximal normal subgroup of	#72280597, #72281079, #72282332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279166);