#154074

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2970 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 2970 conjugacy classes of subgroups — too many to draw. Only its 914 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=i,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72240347, #72240348, C₂×#72240347, …
Fitting subgroup of	#72240347, #72261577, #72261796
maximal subgroup of	#72233925, #72233928, #72240290, …
maximal normal subgroup of	#72233925, #72233928, #72240290, …
maximal quotient of	C₂×#154074, #72257483, #72265565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61270);