#7627358

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×#3539	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7168 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)(7,10)(8,11)(9,12)(13,14,20,18)(15,16,21,17)(19,22,27,26)(23,24,28,25);(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,23)(20,21)(22,25)(24,26)(27,28);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,16,20,17)(14,15,18,21)(19,24,27,25)(22,23,26,28);(13,17,20,16)(14,21,18,15)(19,25,27,24)(22,28,26,23);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(13,18,20,14)(15,16,21,17)(19,22,27,26)(23,25,28,24);(5,6)(7,9)(8,11)(10,12)(13,19,20,27)(14,22,18,26)(15,23,21,28)(16,24,17,25)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=i,f^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, 7534554);