#204959

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
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	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 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)(5,6,11,8)(7,14,18,20)(9,13,10,19)(12,16,15,17)(21,22,25,28,24,27,31,33)(23,26,30,35,29,34,32,36);(5,7,10,17,11,18,9,16)(6,12,13,14,8,15,19,20)(21,23,25,32,24,29,31,30)(22,26,33,35,27,34,28,36);(1,2,3,4)(5,8,11,6)(7,15,18,12)(9,13,10,19)(14,17,20,16)(22,27)(26,34)(28,33)(35,36);(5,9,11,10)(6,13,8,19)(7,16,18,17)(12,14,15,20)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,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, 112155);