#7626336

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×#3554	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)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,23)(17,25)(18,20)(19,26)(22,24)(27,28);(7,11)(9,12)(13,14,18,20)(15,21,23,16)(17,19,26,25)(22,27,28,24);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,22)(18,23)(19,24)(20,21)(25,27)(26,28);(1,2,4,3)(13,16)(14,15)(17,24)(18,21)(19,22)(20,23)(25,28)(26,27);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,17,18,26)(14,19,20,25)(15,22,23,28)(16,24,21,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 7533532);