#377923

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=fh,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,h^c=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, 285119);