#7627220

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3481	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7440 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)(7,10)(8,11)(9,12)(14,16)(15,20)(21,24)(23,28);(5,6)(7,10)(8,11)(9,12)(13,14,19,16)(15,17,20,22)(18,21,27,24)(23,25,28,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,22)(18,23)(19,20)(21,25)(24,26)(27,28);(13,16,19,14)(15,22,20,17)(18,24,27,21)(23,26,28,25);(5,7)(6,10)(8,12)(9,11)(13,17,19,22)(14,20,16,15)(18,26,27,25)(21,23,24,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,28)(22,26)

Polycyclic presentation:

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

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, 7534416);