#2109408

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4346 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,e^2=f,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	#72232286, #72232308, #72232339, …
maximal normal subgroup of	#72232286, #72232308, #72232339, …
maximal quotient of	#72330907, C₂×#2109408

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2016604);