#360194

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240851, #72240857, #72240860, …
maximal normal subgroup of	#72240851, #72240857, #72240860, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267390);