#249056

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2006 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 2006 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156252);