#7626233

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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	C₂²×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8514 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)(6,9)(10,12)(13,14,19,17)(15,22,21,18)(16,24,26,20)(23,25,27,28);(7,11)(10,12)(14,17)(15,21)(20,24)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,22,21,18)(16,24,26,20)(23,28,27,25);(5,6)(7,10)(8,9)(11,12)(13,18,19,22)(14,21,17,15)(16,25,26,28)(20,27,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314984, #72314991, #72315000, …
maximal normal subgroup of	#72314984, #72314991, #72315000, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533429);