#155665

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

5250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250268, #72250278, #72250293, …
maximal normal subgroup of	#72250268, #72250278, #72250293, …
maximal quotient of	#72256502, #72256531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62861);