#205026

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1156 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)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18)(21,22)(23,26,30,33)(24,31)(25,27)(28,32)(29,35,36,34);(6,11,14,13)(12,18,19,20)(21,23)(22,26,27,33)(24,29)(25,30)(28,34,31,35)(32,36);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19)(22,27)(24,32)(26,33)(29,36);(7,15)(12,19)(16,17)(18,20)(21,24,25,32)(22,28,27,31)(23,29,30,36)(26,34,33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fi,d^2=i,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

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, 112222);