#363504

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,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, 270700);