#7626762

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533958);