#7627494

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

8260 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)(13,14,19,20)(15,23,26,21)(16,25,27,17)(18,22,28,24);(6,9)(10,11)(14,20)(16,27)(21,23)(22,24);(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,21,20,23)(16,24,27,22)(17,18,25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,24)(18,21)(19,27)(20,25)(22,26)(23,28);(13,17)(14,16)(15,18)(19,25)(20,27)(21,24)(22,23)(26,28);(13,18)(14,22)(15,25)(16,21)(17,26)(19,28)(20,24)(23,27)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 7534690);