#108145

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh

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, 15341);