#358424

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252691, #72256876, #72256910, …
maximal normal subgroup of	#72252691, #72256876, #72256910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265620);