#185101

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3272 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 3272 conjugacy classes of subgroups — too many to draw. Only its 338 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)(5,6)(7,9)(8,10)(11,12)(14,19,21,25)(20,26,28,27);(6,9)(7,12)(10,11)(13,14)(15,20)(16,19)(17,25)(18,21)(22,27)(23,28)(24,26);(2,4)(13,15,18,23)(14,20,21,28)(16,24,17,22)(19,26,25,27);(6,10)(9,11)(13,16,18,17)(14,19,21,25)(15,22,23,24)(20,27,28,26)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92297);