#1361957

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3137 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326419, #72326421, #72326440
maximal normal subgroup of	#72326419, #72326421, #72326440

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1269153);