#874350

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fg,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256749, #72256754, #72256782
maximal normal subgroup of	#72256749, #72256754, #72256782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 781546);