#453993

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2149 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)(5,6,11,8)(7,15,19,30)(9,13,24,20)(10,21,27,14)(12,25,16,18)(17,33,35,31)(22,26,29,23)(28,36,32,34);(5,7)(6,12)(8,16)(9,17)(10,25)(11,19)(13,28)(14,30)(15,21)(18,27)(20,32)(22,33)(23,36)(24,35)(26,34)(29,31);(2,4)(5,8,11,6)(7,16,19,12)(9,22,24,29)(10,14,27,21)(13,23,20,26)(15,25,30,18)(17,31,35,33)(28,34,32,36);(5,9)(6,13)(7,17)(8,20)(10,26)(11,24)(12,28)(14,22)(15,31)(16,32)(18,36)(19,35)(21,29)(23,27)(25,34)(30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361189);