#2319859

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3286 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327507, #72327509, #72327562
maximal normal subgroup of	#72327507, #72327509, #72327562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2227055);