#188515

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,f^2=i,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, 95711);