#7626675

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×#3485	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 7533871);