#188688

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

934 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)(5,6,11,16,10,15,9,14)(7,13,17,20,8,12,18,19)(21,22,26,23)(24,31,33,36)(25,30,35,28)(27,34,29,32);(1,2,3,4)(5,7,10,8)(6,12,15,13)(9,18,11,17)(14,20,16,19)(21,23,26,22)(24,29,33,27)(25,30,35,28)(31,32,36,34);(5,8,10,7)(6,13,15,12)(9,18,11,17)(14,20,16,19)(21,24,26,33)(22,27,23,29)(25,34,35,32)(28,36,30,31);(5,9,10,11)(6,14,15,16)(7,17,8,18)(12,19,13,20)(22,23)(27,29)(28,30)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=h,d^2=hi,f^2=i,g^2=hi,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, 95884);