#395632

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302828);