#2039922

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fgh,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300384, #72300387, #72300456
maximal normal subgroup of	#72300384, #72300387, #72300456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1947118);