#1186853

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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2974 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)(5,6)(7,8)(9,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22,26,24)(23,28,27,25);(6,10)(8,11)(13,15)(14,18)(16,19)(17,20);(1,2,4,3)(6,10)(7,12)(13,15,17,20)(14,18,19,16)(21,23)(22,25)(24,28)(26,27);(1,2,4,3)(5,7,9,12)(6,11,10,8)(13,15)(14,16)(17,20)(18,19)(21,24,26,22)(23,28,27,25);(1,3,4,2)(5,8,9,11)(6,12,10,7)(13,16)(14,15)(17,18)(19,20)(21,25,26,28)(22,27,24,23)

Polycyclic presentation:

2^9|a^2=gi,c^2=fi,d^2=gh,e^2=gh,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	#72331983, #72332008, #72332042
maximal normal subgroup of	#72331983, #72332008, #72332042
maximal quotient of	#72332975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1094049);