#366455

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

958 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)(6,12)(7,17)(8,21)(9,10)(14,29)(15,16)(18,27)(19,33)(20,24)(22,23)(26,32)(35,36);(1,3,4,2)(5,6,10,15,11,16,9,12)(7,18,17,28,20,27,24,13)(8,14,22,30,23,29,21,25)(19,32,31,36,33,35,34,26);(5,7)(6,13)(8,19)(9,17)(10,24)(11,20)(12,18)(14,26)(15,27)(16,28)(21,31)(22,34)(23,33)(25,32)(29,36)(30,35);(5,8,10,22,11,23,9,21)(6,14,15,30,16,29,12,25)(7,19,24,34,20,33,17,31)(13,26,27,35,28,36,18,32)

Polycyclic presentation:

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

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, 273651);