#423195

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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)(6,11)(7,15)(9,21)(13,26)(14,22)(16,32)(17,23)(18,30)(19,24)(25,28)(27,36)(31,34);(5,6,10,14)(7,12,17,29)(8,18,20,26)(9,22,21,11)(13,24,30,19)(15,25,23,28)(16,33,34,35)(27,31,36,32);(1,2,3,4)(5,7,10,17)(6,12,14,29)(8,16,20,34)(9,23,21,15)(11,25,22,28)(13,27,30,36)(18,35,26,33)(19,31,24,32);(5,8)(6,13)(7,16)(9,24)(10,20)(11,26)(12,27)(14,30)(15,31)(17,34)(18,22)(19,21)(23,32)(25,33)(28,35)(29,36)

Polycyclic presentation:

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

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, 330391);