#428862

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336058);