#356623

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

1511 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:

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

Polycyclic presentation:

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

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, 263819);