#363229

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#3505	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261562
maximal normal subgroup of	#72261562
maximal quotient of	#72261683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270425);