#493374

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3458 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260453, #72261217, #72261251, …
maximal normal subgroup of	#72260453, #72261217, #72261251, …
maximal quotient of	#72260460

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400570);