#392665

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₄×#120
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72343893, #72343905, #72343907, …
maximal subgroup of	#72250213, #72257146, #72257185, …
maximal normal subgroup of	#72250213, #72257146, #72257185, …
maximal quotient of	#72317391, #72317394, C₂×#392665, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299861);