#196481

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1422 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

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, 103677);