#25191

order	256
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#3012	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,17,12,18,11,16)(7,19,23,33,24,32,22,13)(8,25,28,15,9,29,27,14)(20,30,21,31,36,35,26,34);(1,2)(3,4)(5,7,10,23,12,24,11,22)(6,13,17,19,18,33,16,32)(8,26,28,20,9,21,27,36)(14,34,25,30,15,31,29,35);(2,4)(5,8,12,9)(6,14,18,15)(7,20,24,36)(10,27,11,28)(13,30,33,35)(16,25,17,29)(19,34,32,31)(21,23,26,22);(5,9,12,8)(6,15,18,14)(7,21,24,26)(10,27,11,28)(13,31,33,34)(16,25,17,29)(19,35,32,30)(20,23,36,22)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fh,g^c=gh,e^d=eh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25191, #6439019, #6439025, …
maximal normal subgroup of	C₂×#25191, #6439019, #6439025, …
maximal quotient of	#283584, #283586, #283592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18179);