#376902

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 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)(5,6,11,17,10,15,12,16)(7,18,8,21,20,13,23,29)(9,24,26,30,25,14,27,31)(19,28,22,35,33,32,36,34);(2,4)(5,7)(6,13)(8,11)(9,19)(10,20)(12,23)(14,28)(15,18)(16,21)(17,29)(22,26)(24,32)(25,33)(27,36)(30,34)(31,35);(2,4)(5,8,12,7,10,23,11,20)(6,13,17,29,15,18,16,21)(9,22,27,19,25,36,26,33)(14,28,31,35,24,32,30,34);(2,4)(5,9,11,26,10,25,12,27)(6,14,17,31,15,24,16,30)(7,19,8,22,20,33,23,36)(13,28,29,35,18,32,21,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284098);