#1311419

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326971, #72327029, #72327037, …
maximal normal subgroup of	#72326971, #72327029, #72327037, …
maximal quotient of	#72330485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1218615);