#7627177

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×#3541	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6926 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)(13,14,19,17)(15,20,26,25)(16,22,27,21)(18,28,24,23);(1,2,4,3)(6,9)(10,11)(13,15,19,26)(14,20,17,25)(16,24,27,18)(21,28,22,23);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,24)(17,22)(18,26)(19,27)(20,28)(23,25);(5,7)(6,10)(8,12)(9,11)(13,17,19,14)(15,25,26,20)(16,22,27,21)(18,28,24,23);(1,3,4,2)(13,16)(14,22)(15,24)(17,21)(18,26)(19,27)(20,23)(25,28);(13,18,19,24)(14,23,17,28)(15,16,26,27)(20,21,25,22)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=hi,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, 7534373);