#361667

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62812	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3624 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:

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

Polycyclic presentation:

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

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, 268863);