#7627457

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3519	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7584 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534653);