#7626339

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3547	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7052 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)(5,6,11,9)(7,8,12,10)(15,21)(19,22)(23,27)(26,28);(6,9)(7,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26);(5,7)(6,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(1,2,4,3)(5,8,11,10)(6,7,9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26);(1,3,4,2)(5,9,11,6)(7,8,12,10)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28);(5,10,11,8)(6,12,9,7)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533535);