#395584

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2213 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302780);