#1361951

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×#2670	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3095 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)(13,14,18,16)(15,20,17,19)(22,25)(23,27);(5,6,9,8)(7,10,11,12)(13,15,18,17)(14,19,16,20)(21,22)(23,26)(24,25)(27,28);(1,2,4,3)(13,14,18,16)(15,20,17,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,20,17,19)(21,23,24,27)(22,26,25,28);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,17,18,15)(14,19,16,20)(21,22)(23,28)(24,25)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fg,d^2=f,e^2=ghi,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	#72327004, #72327029, #72327049
maximal normal subgroup of	#72327004, #72327029, #72327049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1269147);