#364741

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261691, #72262061, #72262091
maximal normal subgroup of	#72261691, #72262061, #72262091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271937);