#1269010

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3882 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)(5,6,9,12)(7,8,11,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,20)(18,19)(21,22,25,28)(23,24,27,26);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,28)(25,27);(5,7)(6,8)(9,11)(10,12)(13,14)(15,17)(16,19)(18,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,8)(6,11)(7,12)(9,10)(13,15)(14,18)(16,20)(17,19)(21,24)(22,27)(23,28)(25,26)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,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	#72336541, #72336560, #72336612
maximal normal subgroup of	#72336541, #72336560, #72336612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1176206);