#6527054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28337	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434250);