#6457340

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2444 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)(5,6,8,10)(7,9,11,12)(13,14,19,23)(15,24,25,16)(17,22,18,28)(20,21,27,26);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,25,17,21)(14,20,22,24,23,27,28,16);(7,11)(9,12)(13,16,18,20,19,24,17,27)(14,21,22,15,23,26,28,25);(7,11)(9,12)(15,25)(16,24)(20,27)(21,26);(13,17,19,18)(14,22,23,28)(15,26,25,21)(16,27,24,20)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,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^b=ei,d^c=di

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, 6364536);