#361632

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3568 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)(7,15,18,11)(8,19,21,28)(9,14)(10,13)(12,20,26,22)(16,31)(17,30,35,25)(23,27)(24,32)(29,34)(33,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,36)(20,32)(21,33)(22,34)(23,35)(26,29)(27,30)(28,31);(2,4)(5,8,10,21)(6,12,13,26)(7,16,18,33)(9,22,23,20)(11,24,15,29)(14,28,27,19)(17,34,35,32)(25,31,30,36);(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,36)(20,32)(21,33)(22,34)(23,35)(26,29)(27,30)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,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, 268828);