#386365

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 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)(6,12,15,23)(7,16,18,24)(14,19,30,26)(17,32,34,31);(5,6)(7,13)(8,19)(9,23)(10,12)(11,15)(14,20)(16,28)(17,33)(18,29)(21,30)(22,26)(24,25)(27,32)(31,36)(34,35);(5,7,11,18)(6,13)(8,17,22,34)(9,16,10,24)(12,25)(14,27)(15,29)(19,33)(20,32,21,31)(23,28)(26,35)(30,36);(5,8,11,22)(6,14,15,30)(7,17,18,34)(9,21,10,20)(12,26,23,19)(13,27,29,36)(16,31,24,32)(25,35,28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293561);