#367097

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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)(7,16)(8,18)(9,23)(10,26)(11,15)(12,21)(13,24)(14,27)(17,31)(19,22)(20,28)(25,35)(29,36)(30,32)(33,34);(5,7,11,12)(6,8)(9,17,25,28)(10,18,27,19)(13,20)(14,16)(15,22)(21,26)(23,30)(24,32,36,33)(29,31)(34,35);(2,4)(5,8)(6,12)(7,15)(9,20)(10,16)(11,22)(13,28)(14,19)(17,29)(18,26)(21,27)(23,33)(24,30)(25,31)(32,35)(34,36);(5,9,10,24,11,25,27,36)(6,13,14,23,15,29,26,35)(7,17,18,32,12,28,19,33)(8,20,16,30,22,31,21,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274293);