#188584

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
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

904 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)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19)(21,22,26,30,27,29,25,35)(23,28,33,36,31,24,32,34);(1,2,3,4)(21,23)(22,28)(24,29)(25,32)(26,33)(27,31)(30,36)(34,35);(5,7)(6,10)(8,16)(9,15)(11,18)(12,20)(13,19)(14,17)(21,24,27,28)(22,23,29,31)(25,36,26,34)(30,33,35,32);(6,11)(10,18)(12,19)(13,20)(22,29)(23,31)(30,35)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95780);