#392140

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14679
prime radical	#63103	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

15739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341447, #72341458, #72341463
maximal subgroup of	#72215925, #72219682, #72219688, …
maximal normal subgroup of	#72215925, #72219682, #72219688, …
maximal quotient of	#72234547, #72234936, #72234937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299336);