#846486

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4578 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)(6,7)(8,12)(13,14)(15,19)(16,20)(17,18)(21,22,24,26)(23,27,28,25);(5,6,9,7)(8,10,12,11)(14,18)(15,20)(21,23,24,28)(22,25,26,27);(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,23)(22,25)(24,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331733, #72331739, #72331756, …
maximal normal subgroup of	#72331733, #72331739, #72331756, …
maximal quotient of	#72332521, #72332523

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 753682);