#377649

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284845);