#7604312

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45042	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=hi,d^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7511508);