#367935

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1103 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)(7,11,17,27)(8,13,19,30)(9,15)(10,14)(12,18,26,21)(16,32)(20,29,22,31)(23,24)(25,34)(28,35)(33,36);(1,2,3,4)(5,7)(6,11,14,27)(8,16)(9,21)(10,17)(12,24,26,15)(13,25,30,32)(18,23)(19,34)(20,35)(22,33)(28,29,36,31);(2,4)(5,7)(6,12)(8,16)(9,21)(10,17)(11,24)(13,28)(14,26)(15,27)(18,23)(19,34)(20,35)(22,33)(25,29)(30,36)(31,32);(5,8,9,22,10,19,23,20)(6,13,24,29,14,30,15,31)(7,16,21,33,17,34,18,35)(11,25,26,36,27,32,12,28)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275131);