#422308

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1155 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)(3,4)(5,6,10,15,11,16,27,26)(7,17,19,29,20,9,14,12)(8,21,24,33,25,32,23,13)(18,28,31,22,36,30,35,34);(5,7,11,20)(6,12,16,29)(8,18)(9,26,17,15)(10,14,27,19)(13,30)(21,22)(23,31)(24,35)(25,36)(28,33)(32,34);(5,8)(6,13)(7,18)(9,22)(10,23)(11,25)(12,28)(14,31)(15,32)(16,33)(17,34)(19,35)(20,36)(21,26)(24,27)(29,30);(2,4)(5,9,10,12,11,17,27,29)(6,14,15,7,16,19,26,20)(8,22,23,28,25,34,24,30)(13,31,32,18,33,35,21,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329504);