#767094

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3632 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)(5,6)(7,11)(8,10)(9,12)(14,17)(15,19);(5,6,8,10)(7,9,12,11)(13,14)(15,20)(16,17)(18,19)(22,25)(26,28);(1,2)(3,4)(7,12)(9,11)(14,17)(18,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,26,25,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=h,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	#72331739, #72331745, #72331766
maximal normal subgroup of	#72331739, #72331745, #72331766
maximal quotient of	#72333139

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 674290);