#249054

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2135 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 2135 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227754, #72227761, #72227769
maximal normal subgroup of	#72227754, #72227761, #72227769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156250);