#1295725

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3070 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)(7,12)(8,11)(9,10)(13,14)(15,16)(17,18)(19,20)(21,22)(23,27)(24,25)(26,28);(5,7,10,8)(6,11,9,12)(13,15,19,18)(14,17,20,16)(21,23)(22,26)(24,27)(25,28);(5,7,10,8)(6,11,9,12)(13,16,19,17)(14,15,20,18)(21,22,25,24)(23,27,28,26);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,17,19,16)(14,15,20,18)(21,23)(22,27)(24,26)(25,28);(5,9,10,6)(7,12,8,11)(13,18,19,15)(14,17,20,16)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|b^2=gh,c^2=f,d^2=fhi,e^2=f,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1202921);