#423434

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1331 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)(7,11,17,24)(8,12)(9,14)(10,13)(15,29,31,18)(16,23,33,36)(19,26)(20,25)(21,28)(22,27)(30,34,32,35);(5,6)(7,15,17,31)(8,12)(9,14)(10,13)(11,18,24,29)(16,32,33,30)(19,26)(20,25)(21,28)(22,27)(23,34,36,35);(5,7)(6,11,13,24)(8,16)(9,18)(10,17)(12,23,25,36)(14,15,27,31)(19,34)(20,33)(21,35)(22,29)(26,32,28,30);(2,4)(5,8,10,20)(6,12,13,25)(7,16,17,33)(9,21,22,19)(11,23,24,36)(14,28,27,26)(15,30,31,32)(18,35,29,34)

Polycyclic presentation:

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

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, 330630);