#386967

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fg,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215017, #72216146, #72217006
maximal normal subgroup of	#72215017, #72216146, #72217006
maximal quotient of	#72214933

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294163);