#374043

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 281239);