#1385998

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	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3318 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)(5,6,9,11)(7,8,12,10)(13,14,17,19)(15,20,16,18)(23,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,18)(17,19);(1,2)(3,4)(13,15,17,16)(14,18,19,20)(21,22,25,24)(23,26,27,28);(5,8,9,10)(6,7,11,12)(13,16,17,15)(14,18,19,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,20)(17,19)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=fgh,c^2=fi,d^2=f,e^2=fhi,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	#72297818, #72297819, #72297859
maximal normal subgroup of	#72297818, #72297819, #72297859
maximal quotient of	#72298558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1293194);