#188657

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95853);