#6463217

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370413);