#358031

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1814 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)(7,16)(8,21,24,35)(9,25)(10,14)(11,15)(12,19)(13,23,28,22)(17,31,32,26)(18,30)(20,27)(29,34,36,33);(1,2)(3,4)(5,7,9,19,11,18,10,20)(6,12,14,16,15,27,25,30)(8,17,22,33,24,32,23,34)(13,26,21,29,28,31,35,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,19,33,18,32,20,34)(12,26,16,29,27,31,30,36);(7,18)(12,27)(16,30)(17,32)(19,20)(26,31)(29,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237649, #72237658, #72237671, …
maximal normal subgroup of	#72237649, #72237658, #72237671, …
maximal quotient of	#72314700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265227);