#366454

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=e,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, 273650);