#7627384

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×#3554	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=gi,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, 7534580);