#7626660

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	C₂×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6906 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=g,f^2=gh,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, 7533856);