#365126

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 272322);