#188698

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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)(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(1,2,3,4)(21,22)(23,26)(24,28)(25,27)(29,30)(31,35)(32,34)(33,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(6,11)(10,13)(12,19)(18,20)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

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

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, 95894);