#365459

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#575	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,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,e^b=ei,f^b=fi,g^c=gi,e^d=ei

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, 272655);