#1573090

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4021 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)(5,6,8,9)(7,11,12,10)(13,14,17,15)(16,19,20,18)(21,22)(23,27)(24,26)(25,28);(6,9)(7,12)(13,14)(15,17)(16,19)(18,20)(21,22,24,26)(23,25,28,27);(6,9)(10,11)(14,15)(18,19)(23,28)(25,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23,24,28)(22,25,26,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,18)(15,19)(17,20)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326390, #72326403, #72326432, …
maximal normal subgroup of	#72326390, #72326403, #72326432, …
maximal quotient of	#72328844, #72333312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1480286);