#996703

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3094 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)(5,6,9,10)(7,11,12,8)(13,14,18,16)(15,17,20,19)(22,26)(24,28);(5,7,9,12)(6,8,10,11)(13,15,18,20)(14,19,16,17)(21,22,25,26)(23,24,27,28);(6,10)(7,12)(13,14,18,16)(15,17,20,19)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,16,18,14)(15,17,20,19);(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,27)(23,26)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331959, #72331962, #72332057
maximal normal subgroup of	#72331959, #72331962, #72332057
maximal quotient of	#72333107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 903899);