#7627387

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×#3551	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7204 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(1,2,4,3)(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7,9,8)(6,10,11,12)(13,16,17,24)(14,19,21,27)(15,22,23,28)(18,25,20,26);(1,3,4,2)(5,6)(7,10)(8,12)(9,11)(13,15)(14,20)(16,22)(17,23)(18,21)(19,26)(24,28)(25,27);(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,19)(17,21)(20,23)(22,25)(24,27)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,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, 7534583);