#361226

order	512
nilpotency class	3
isoclinism	isoclinic class of #359901
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2342
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13892
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339024, #72339028, #72339037
maximal normal subgroup of	#72339024, #72339028, #72339037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268422);