#356506

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1872 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263702);