#1105416

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3482 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=gh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323459, #72323460, #72323463
maximal normal subgroup of	#72323459, #72323460, #72323463
maximal quotient of	#72323532, #72323550

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1012612);