#373684

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2169 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)(5,6)(7,17)(8,21)(9,15)(10,13)(11,16)(12,19)(14,22)(18,26)(20,28)(23,30)(24,29)(25,27)(31,32)(33,36)(34,35);(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,28)(14,26)(15,17)(16,27)(21,31)(22,34)(23,32)(24,33)(29,35)(30,36);(1,3,4,2)(6,13)(9,10)(12,17)(14,29)(15,16)(19,25)(21,30)(22,23)(26,31)(27,28)(32,34)(35,36);(5,8,11,24)(6,14,16,30)(7,18,20,33)(9,23,10,22)(12,26,27,36)(13,29,15,21)(17,31,28,35)(19,34,25,32)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,e^d=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, 280880);