#395428

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3507 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)(7,15,18,11)(8,20,21,12)(9,14)(10,13)(16,33)(17,34)(19,31,23,28)(22,32,24,29)(25,30)(26,35)(27,36);(5,7)(6,11)(8,17)(9,23)(10,18)(12,27)(13,15)(14,31)(16,22)(19,25)(20,34)(21,36)(24,35)(26,29)(28,30)(32,33);(2,4)(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,20)(14,32)(15,33)(17,23)(18,35)(19,36)(22,25)(27,31)(28,34)(29,30);(5,8)(6,12)(7,17)(9,22)(10,21)(11,27)(13,20)(14,29)(15,34)(16,23)(18,36)(19,35)(24,25)(26,31)(28,33)(30,32)

Polycyclic presentation:

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

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, 302624);