#361712

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268908);