#358864

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
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×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3066 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 3066 conjugacy classes of subgroups — too many to draw. Only its 378 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261540
maximal normal subgroup of	#72261540
maximal quotient of	#72261587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266060);