#366713

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2240 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)(7,17)(8,22,26,31)(9,15)(10,14)(11,16)(12,19)(13,24,32,25)(18,34,36,27)(20,29)(21,28)(23,33,35,30);(1,2,3,4)(5,7)(6,12)(8,23,26,35)(9,20)(10,19)(11,21)(13,30,32,33)(14,28)(15,17)(16,29)(18,25,36,24)(22,27,31,34);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,33)(19,23)(20,35)(21,36)(28,30)(29,34);(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,27)(14,28)(15,17)(16,29)(22,33)(23,25)(24,35)(26,36)(30,31)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273909);