#1427653

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

3892 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)(7,8)(9,10)(11,12)(13,14,17,19)(15,20,18,16)(21,22)(23,28)(24,26)(25,27);(6,10)(8,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28);(5,7)(6,11)(8,10)(9,12)(13,16)(14,18)(15,19)(17,20)(21,24,25,28)(22,26,27,23);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,26)(27,28);(5,8)(6,12)(7,10)(9,11)(13,14)(15,20)(16,18)(17,19)(21,22)(23,28)(24,26)(25,27)

Polycyclic presentation:

2^9|a^2=fghi,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326406, #72326410, #72326413
maximal normal subgroup of	#72326406, #72326410, #72326413

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1334849);