#392902

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,g^b=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285816, D₄×#1868, #72342938, …
maximal normal subgroup of	#72285816, D₄×#1868, #72342938, …
maximal quotient of	#72285976, #72343149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300098);