#6460028

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	#35563	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6367224);