#769886

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4354 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)(5,6,7,8)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28);(9,10,11,12)(14,16)(19,21)(20,24)(27,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,19)(16,21)(17,25)(18,23)(20,28)(22,26)(24,27);(1,2,3,4)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27);(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330803, #72330818, #72330857, …
maximal normal subgroup of	#72330803, #72330818, #72330857, …
maximal quotient of	#72332519

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 677082);