#476996

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384192);