#377933

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=fh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,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, 285129);