#356231

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 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)(7,16,19,12)(8,13)(9,14)(10,15)(11,18,26,24)(17,28,32,27)(20,33,34,25)(21,29)(22,30)(23,31)(35,36);(1,2,3,4)(5,7)(6,11)(8,20)(9,24)(10,19)(12,14)(13,28)(15,26)(16,29)(17,22)(18,21)(23,34)(25,30)(27,31)(32,35)(33,36);(5,7)(6,12)(8,17)(9,18)(10,19)(11,14)(13,27)(15,16)(20,22)(21,24)(23,32)(25,30)(26,29)(28,31)(33,36)(34,35);(5,8,21,35,10,23,9,22)(6,13,29,36,15,31,14,30)(7,17,24,34,19,32,18,20)(11,25,12,27,26,33,16,28)

Polycyclic presentation:

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

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, 263427);