#358421

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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)(7,17,20,28)(8,13,24,32)(9,14)(10,15)(11,16)(12,19,29,21)(18,30)(22,33)(23,31,25,27)(26,34)(35,36);(1,2)(3,4)(5,7,11,20)(6,12)(8,22,24,9)(10,19,26,21)(13,30)(14,27)(15,17)(16,29)(18,23,35,25)(28,34)(31,33)(32,36);(2,4)(5,8)(6,13)(7,18)(9,21)(10,23)(11,24)(12,14)(15,31)(16,32)(17,30)(19,22)(20,35)(25,26)(27,34)(28,36)(29,33);(5,9,10,18,11,22,26,35)(6,14,15,30,16,33,34,36)(7,8,19,23,20,24,21,25)(12,27,17,13,29,31,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250386, #72252967, #72256910, …
maximal normal subgroup of	#72250386, #72252967, #72256910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265617);